The Schur functor on tensor powers

Abstract

Let M be a left module for the Schur algebra S(n, r), and let s ∈ Z + . Then M ⊗s is a (S(n, rs), F Ss)-bimodule, where the symmetric group Ss on s letters acts on the right by place permutations. We show that the Schur functor frs sends M ⊗s to the (F Srs, F Ss)-bimodule F Srs ⊗ F (Sr Ss) ((frM ) ⊗s ⊗F F Ss). As a corollary, we obtain the image under the Schur functor of the Lie power L s (M ), exterior power /\ s (M ) of M and symmetric power S s (M ). Mathematics Subject Classification (2010). Primary 20G43, 20C30.

Vertices of Lie modules

“…The next theorem concerning decompositions of Lie modules is a consequence of the results in [3], [6] and [24]. This decomposition and Theorem 5.4 will then enable us to reduce the problem of determining vertices of indecomposable direct summands of Lie pf F (n) to the case where n is a p-power.…”

“…One key ingredient of our approach is a decomposition theorem, expressing Lie F (n) as a direct sum of pieces related to Lie modules Lie F (p d ), for various d such that p d divides n. This is obtained by translating the Bryant-Schocker decomposition theorem [6] for Lie powers to Lie modules, using work of Lim-Tan [24]. This paves the way to reduce questions on Lie modules to the case when n is a power of p, and puts the Lie modules Lie F (p d ) into the focus of study.…”

The Complexity of the Lie Module

“…When n s, the Schur functor f s sends V ⊗s to the (FS s , FS s )-bimodule FS s , and it sends L s (V ) to the left FS s -module Lie(s). The effect of the Schur functor f rs on V ⊗s and L s (V ), when V is a left S F (n, r)-module and n rs, is described in detail in [10]. In this paper, we need the latter result.…”

“…We make use of this theorem. By work in [10], it may be transferred to the context of symmetric groups. Our main results (Theorems 3.2 and 3.3) show that the complexity of Lie(n) is bounded above by m, where p m is the largest p-power dividing n, and, if n is not a p-power, is equal to the maximum of the complexities of Lie(p i ) with 1 i m. We conjecture that our upper bound is in fact an equality, and show that this conjecture is equivalent to the assertion that the complexity of Lie(p m ) as an FE m -module is m, where E m is a regular elementary abelian subgroup of S p m of order p m .…”

Block components of the Lie module for the symmetric group

“…We regard Lie(k) as an F S * k -module by means of the isomorphism α → α * from S k to S * k . Then Lie(k) may also be regarded as an F S Lemma 2.3 [Lim and Tan 2012]. In the above notation,…”